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Theorem funbrfv 5460
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
funbrfv  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )

Proof of Theorem funbrfv
StepHypRef Expression
1 funrel 5176 . . . 4  |-  ( Fun 
F  ->  Rel  F )
2 brrelex2 4681 . . . 4  |-  ( ( Rel  F  /\  A F B )  ->  B  e.  _V )
31, 2sylan 459 . . 3  |-  ( ( Fun  F  /\  A F B )  ->  B  e.  _V )
4 breq2 3967 . . . . . 6  |-  ( y  =  B  ->  ( A F y  <->  A F B ) )
54anbi2d 687 . . . . 5  |-  ( y  =  B  ->  (
( Fun  F  /\  A F y )  <->  ( Fun  F  /\  A F B ) ) )
6 eqeq2 2265 . . . . 5  |-  ( y  =  B  ->  (
( F `  A
)  =  y  <->  ( F `  A )  =  B ) )
75, 6imbi12d 313 . . . 4  |-  ( y  =  B  ->  (
( ( Fun  F  /\  A F y )  ->  ( F `  A )  =  y )  <->  ( ( Fun 
F  /\  A F B )  ->  ( F `  A )  =  B ) ) )
8 funeu 5182 . . . . . 6  |-  ( ( Fun  F  /\  A F y )  ->  E! y  A F
y )
9 tz6.12-1 5442 . . . . . 6  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
108, 9sylan2 462 . . . . 5  |-  ( ( A F y  /\  ( Fun  F  /\  A F y ) )  ->  ( F `  A )  =  y )
1110anabss7 797 . . . 4  |-  ( ( Fun  F  /\  A F y )  -> 
( F `  A
)  =  y )
127, 11vtoclg 2794 . . 3  |-  ( B  e.  _V  ->  (
( Fun  F  /\  A F B )  -> 
( F `  A
)  =  B ) )
133, 12mpcom 34 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( F `  A )  =  B )
1413ex 425 1  |-  ( Fun 
F  ->  ( A F B  ->  ( F `
 A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   E!weu 2117   _Vcvv 2740   class class class wbr 3963   Rel wrel 4631   Fun wfun 4632   ` cfv 4638
This theorem is referenced by:  funopfv  5461  fnbrfvb  5462  fvelima  5473  fvi  5478  fmptco  5590  fliftfun  5710  fliftval  5714  opabiota  6224  fpwwe2  8198  nqerid  8490  sum0  12124  sumz  12125  fsumsers  12131  isumclim  12150  dvadd  19216  dvmul  19217  dvco  19223  dvcj  19226  dvrec  19231  dvcnv  19251  dvef  19254  ftc1cn  19317  ulmdv  19707  minvecolem4b  21382  minvecolem4  21384  hlimuni  21743  chscllem4  22162  fvtransport  23995  fvray  24104  fvline  24107
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654
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